Background: Analytical study is presented on the transient problem of buoyancy-induced motion due to the presence of a hot aerosol sphere in unbounded quiescent fluid. Method of Approach: Because the initial flow field is identically zero, the initial stage of the process is governed by viscous and buoyancy forces alone where the convective inertial terms in the momentum and energy balances are negligible, i.e., the initial development of the field is a linear process. The previous statement is examined by analyzing the scales of the various terms in the Navier-Stokes and energy equations. This scale analysis gives qualitative limitations on the validity of the linear approximation. A formal integral solution is obtained for arbitrary Prandtl number and for transient temperature field. Results: We consider, in detail, the idealized case of vanishing Prandtl number for which the thermal field is developed much faster than momentum. In this case, analytical treatment is feasible and explicit expressions for the field variables and the drag acting on the particle are derived. Detailed quantitative analysis of the spatial and temporal validity of the solution is also presented. Conclusions: The linear solution is valid throughout space for $t<10$ diffusion times. For $t>10$, an island in space appears in which inertial effects become dominant. The transient process is characterized by two different time scales: for short times, the development of the field is linear, while for small distances from the sphere and finite times, it is proportional to the square root of time. The resultant drag force acting on the sphere is proportional to the square root of time throughout the process.

1.
Gebhart
,
B.
,
Jaluria
,
Y.
,
Mahajan
,
R.
, and
Sammakia
,
B.
, 1988,
Buoyancy Induced Flows and Transport
,
Hemisphere
, Washington, DC.
2.
Zel’dovich
,
Ya. B.
, 1937, “
Limiting Laws of Freely Rising Convective Currents
,”
Zh. Eksp. Teor. Fiz.
0044-4510,
7
, pp.
1463
1465
(in Russian).
3.
Yih
,
C. S.
, 1953, “
Free Convection Due to Boundary Sources: Fluid Models in Geophysics
,”
Proc. 1st Symposium on Use of Models in Geophysical Fluid Dynamics
,
Government Printing Office
, Washington, DC, pp.
117
133
.
4.
Gutman
,
L. N.
, 1949, “
On Laminar Thermal Convection Above a Stationary Heat Source
,”
Prikl. Mat. Mekh.
0032-8235,
13
, pp.
435
439
(in Russian).
5.
Fujii
,
T.
, 1963, “
Theory of Steady Laminar Convection Above a Horizontal Line Heat Source and a Point Heat Source
,”
Int. J. Heat Mass Transfer
0017-9310,
6
, pp.
597
606
.
6.
Kurdyumov
,
V. N.
, and
Liñán
,
A.
, 1998, “
Free convection From a Point Source of Heat, and Heat Transfer From Spheres at Small Grashof Numbers
,”
Int. J. Heat Mass Transfer
0017-9310,
42
, pp.
3849
3860
.
7.
Chen
,
T. S.
, and
Mucoglu
,
A.
, 1977, “
Analysis of Mixed Forced and Free Convection About a Sphere
,”
Int. J. Heat Mass Transfer
0017-9310,
20
, pp.
867
875
.
8.
Wong
,
K. L.
,
Lee
,
S. C.
, and
Chen
,
C. K.
, 1986, “
Finite Element Solution of Laminar Combined Convection From a Sphere
,”
ASME J. Heat Transfer
0022-1481,
108
, pp.
860
865
.
9.
Cameron
,
M. R.
,
Jeng
,
D. R.
, and
De-Witt
,
K. J.
, 1991, “
Mixed Forced and Natural Convection From Two-Dimensional or Axisymmetric Bodies of Arbitrary Contour
,”
Int. J. Heat Mass Transfer
0017-9310,
34
, pp.
582
587
.
10.
Potter
,
J. M.
, and
Riley
,
N.
, 1980, “
Free Convection From a Heated Sphere at Large Grashof Numbers
,”
J. Fluid Mech.
0022-1120,
100
(
4
), pp.
769
783
.
11.
Geoola
,
F.
, and
Cornish
,
A. R. H.
, 1981, “
Numerical Solution of Steady State Free Convection Heat Transfer From a Solid Sphere
,”
Int. J. Heat Mass Transfer
0017-9310,
24
(
8
), pp.
1369
1379
.
12.
Jia
,
H.
, and
Gogos
,
G.
, 1995, “
Laminar Natural Convection Heat Transfer From Isothermal Sphere
,”
Int. J. Heat Mass Transfer
0017-9310,
39
(
8
), pp.
1603
1615
.
13.
Fendell
,
F. E.
, 1968, “
Laminar Natural Convection About an Isothermally Heated Sphere at Small Grashof Numbers
,”
J. Fluid Mech.
0022-1120,
34
, pp.
163
176
.
14.
Hieber
,
C. A.
, and
Gebhart
,
B.
, 1969, “
Mixed Convection From a Sphere at Small Reynolds and Grashof Numbers
,”
J. Fluid Mech.
0022-1120,
38
, pp.
137
159
.
15.
Abramzon
,
B.
, and
Elata
,
C.
, 1983, “
Unsteady Heat Transfer From a Single Sphere in Stokes Flow
,”
Int. J. Heat Mass Transfer
0017-9310,
27
(
5
), pp.
687
695
.
16.
Feng
,
Z. G.
, and
Michaelides
,
E. E.
, 1998, “
Transient Heat Transfer From a Particle With Arbitrary Shape and Motion
,”
ASME J. Heat Transfer
0022-1481,
120
, pp.
674
681
.
17.
Feng
,
Z. G.
, and
Michaelides
,
E. E.
, 1999, “
A Numerical Study on the Transient Heat Transfer From a Sphere at High Reynolds and Peclet Numbers
,”
Int. J. Heat Mass Transfer
0017-9310,
43
, pp.
219
229
.
18.
Geoola
,
F.
, and
Cornish
,
A. R. H.
, 1982, “
Numerical Simulation of Free Convection Heat Transfer From a Solid Sphere
,”
Int. J. Heat Mass Transfer
0017-9310,
25
, pp.
1677
1687
.
19.
Dudek
,
D. R.
,
Fletcher
,
T. H.
,
Longwell
,
J. P.
, and
Sarofim
,
A. F.
, 1988, “
Natural-Convection Induced Drag Forces on Spheres at Low Grashof Numbers—Comparison of Theory With Experiment
,”
Int. J. Heat Mass Transfer
0017-9310,
31
, pp.
863
873
.
20.
Davis
,
E. J.
, and
Schweiger
,
G.
, 2002,
The Airborne Microparticle, Its Physics, Chemistry, Optics, and Transport Phenomena
,
Springer-Verlag
, Berlin.
21.
Mograbi
,
E.
, and
Bar-Ziv
,
E.
, 2004, “
Dynamics of a Spherical Particle in Mixed Convection Flow Field
,”
J. Aerosol Sci.
0021-8502,
36
(
3
), pp.
387
409
.
22.
Landau
,
L. D.
, and
Lifshitz
,
E. M.
, 1999,
Fluid Mechanics
, 2nd ed.,
Butterworth-Heinemann
, Oxford.
23.
Polyanin
,
A. D.
, 2002,
Handbook of Linear Partial Differential Equations for Engineers and Scientists
,
Chapman & Hall/CRC
, New York, p.
81
.
24.
Hinch
,
E. J.
, 1991,
Perturbation Methods
,
Cambridge University Press
, New York.
25.
Basset
,
A. B.
, 1888,
A Treatise on Hydrodynamics
,
Deighton Bell
, London.
26.
Maxey
,
M. R.
, and
Riley
,
J. J.
, 1983, “
Equation of Motion for a Small Rigid Sphere in a Nonuniform Flow
,”
Phys. Fluids
0031-9171,
26
, pp.
883
889
.
27.
Lovalenti
,
P. M.
, and
,
J. F.
, 1993, “
The Hydrodynamic Force on a Rigid Particle Undergoing Arbitrary Time-Dependent Motion at Small Reynolds Number
,”
J. Fluid Mech.
0022-1120,
256
, pp.
561
605
.
28.
Coimbra
,
C. F. M.
, and
Rangel
,
R. H.
, 1998, “
General Solution of the Particle Momentum Equation in Unsteady Stokes Flows
,”
J. Fluid Mech.
0022-1120,
370
, pp.
53
72
.